The 'tex' tags didn't work for me either so I took them out.W = F * D
I'm not sure if the integral should go from 0 to 10 or 0 to 6
\delta\pi(6-y) * (6-y)dy
I know this is a big mess, and I'm doing this horribly wrong but I just wanted to give my attempts before asking for help.
P.S I don't know what's wrong with these equations
A "layer of water" of height dy has volume Area*height= \pi(5)^2*dx= 25\pi dx and so weight 25\pi \delta dy. It has to be lifted distance 10- y so the work done lifting one "layer" is 25\pi (10- y)dy. Since the water to be lifted goes from y= 0 to y= 6, integrate between 0 and 6.